definitely talk about this with your supervisor. Grieving takes times and as sooner you will make it a priority, the sooner you will be ready to go back to your work. Dont hope that you can only focus on work and postpone grieving...doesnt work...
it is possible to postpone deadline is such cases, your supervisor will know it better how.
(I am talking from my own experience- lost last year a very, very close memmber of family but was back to my phD in the next week after funeral. The result- serious depression and lots of physical problems. do not suggest to do that).
definitely give time for yourself, despite it is not direct loss, for you it is a loss, as one could see from your message. Be nice towards yourself.